Finding intervals where the second derivative is positive 

If f(x)=3x^5−10x^3+5, on what intervals is f′′(x)>0.

Calculus
Shile Shile
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1 Answer

Lets compute $f''(x)$: 

$$f'(x)=15x^4−30x^2.$$
\[f''(x)=60x^3-60x=60x(x^2-1)=60x(x+1)(x-1).\]
The roots of $f''(x)=60x(x+1)(x-1)$ are $x=0, x=1, x=-1$. Hence $f''>0$ on the following intervals 
\[(1,\infty)   \text{and}   (-1,0).\]

We can also see that Hence $f''<0$ on the following intervals 
\[(-\infty,-1)   \text{and}   (0,1).\]

Savionf Savionf
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